In this paper, rank one strange attractor in a periodically kicked time-delayed system is investigated. It is shown that rank one strange attractors occur when the delayed system under a periodic forcing undergoes Hopf bifurcation. Our discussion is based on the theory of rank one maps formulated by Wang and Young. As an example, periodically kicked Chua’s system with time-delay is considered, conditions for rank one chaos along with the results of numerical simulations are presented.

Recently, a chaos theory on rank one maps has been developed by Wang and Young. In 2001, Wang and Oksasoglu [1] gave simple conditions that guarantee the existence of strange attractors with a single direction of instability and certain controlled behaviors. In 2008, Wang and Young accomplished a more comprehensive understanding of the complicated geometric and dynamical structures of a specific class of non-uniformly hyperbolic homoclinic tangles. For certain differential equations, through their well-defined computational process, the existence of the indicated phenomenon of rank one chaos was verified [2]. In 2009, Chen and Han studied the existence of rank one chaos in a periodically kicked planar equation with heteroclinic cycle [3]. In 2012, Fang studied the synchronization between rank-one chaotic systems without and with delay using linear delayed feedback control method [4].

where \(v_{C_{1}}\), \(v_{C_{2}}\), \(i_{L}\) denote voltage across \(C_{1}\), \(C_{2}\) and current through L, respectively. They reported that a chaotic attractor has been observed in Eq. (1). In [6], Wang et al. considered Eq. (1) as the following form:

where \(f(x)\) is chosen to be a cubic function of the form \(f(x) = cx + dx^{3}\). They researched rank one strange attractors in this periodically kicked Chua’s circuit.

In this paper, we first try to develop rank one theory from an ordinary differential equation to a time-delayed system, then consider Chua’s system (3) as the following form:

where a, b, c, d and τ are real parameters, \(\varepsilon{P_{T}}y(t)\) is a time-periodic forcing term with \(P_{T}=\sum_{n=-\infty}^{\infty}\delta(t-nt)\). The stability of equilibria, the bifurcating of periodic solutions and the rank one chaos of the periodically kicked delayed system are investigated.

This paper is organized as follows. In Section 2, we give preliminaries about the rank one chaotic theory. In Section 3, we derive the rank one chaotic theory for a time-delayed system. In Section 4, we take Chua’s system as an example. In Section 5, numerical simulations are presented. Conclusions are given in Section 6.

To properly motivate the studies presented in this paper, we first give a brief overview on the studies of rank one strange attractors, which can be constructed in the following way.

Following the work done by Wang and Young in [7], we let \(\mathbf{u}\in\mathrm{R}^{m}\), \(m\geq2\) be the phase variable and \(t\in\mathrm{R}\) be the time. Consider the following system of equations:

where \(\mathbf{A}_{\mu}\) is a real m by m matrix, and \(f_{\mu}(\mathbf{u})\) is a vector-valued real analytic function in u defined on a given neighborhood of \(\mathbf{u}=\mathbf{0}\) such that \(f\mathbf{(0)}=\mathbf{0}\), \(D_{\mathbf{u}}f\mathbf{(0)}=\mathbf{0}\). Both \(\mathbf{A}_{\mu}\) and \(f_{\mu}(\mathbf{u})\) are smooth dependents of parameter μ around \(\mu=0\). ε is a parameter that controls the magnitude of the forcing, \(\Phi (\mathbf{u})\) is a real vector-valued function of u that represents the shape of the forcing and \(P_{T}=\sum^{\infty}_{n=-\infty}\delta(t-nT)\). When \(\varepsilon=0\), the undisturbed system is

Firstly, we give several definitions.

Definition 1

Let \(f:M\rightarrow M\) be a diffeomorphism of a compact Riemannian manifold onto itself. We say that f is an Anosov diffeomorphism if the tangent space at every \(x \in M\) is split into \(E^{u}(x) \oplus E^{s}(x)\), where \(E^{u}\) and \(E^{s}\) are Df-invariant subspaces, \(Df|_{E^{u}}\) is uniformly expanding and \(Df|_{E^{s}}\) is uniformly contracting. A compact f-invariant set \(\Lambda\subset M\) is called an attractor if there is a neighborhood U of Λ called its basin such that \(f^{n} x \rightarrow \Lambda\) for every \(x \in U\). Λ is called an Axiom A attractor if the tangent bundle over Λ is split into \(E^{u} \oplus E^{s}\) as above.

Definition 2

Let f be a \(C^{2}\) diffeomorphism with an Axiom A attractor Λ. Then there is a unique f-invariant Borel probability measure μ on Λ that is characterized by each of the following (equivalent) conditions:

Further let \(\{{\hat{\mathbf{s}}_{0}=(\cos \theta,\sin\theta,\mathbf{0})\in\mathrm{S}^{m-1},\theta\in [0,2\pi)}\}\) be the unit circle in \((\xi,\eta)\)-plane in \((\xi,\eta,\mathbf{w})\)-space, and define

Theorem 1

Let the values ofμandεbe fixed and assume that (a)-(c) hold. Regard the periodTof the forcing as a parameter and define\(F_{T} = F_{\mu,\varepsilon,T}\). Then there exists a constant \(K_{1}\), determined exclusively by\(\varphi(\theta)\), such that if

$$\biggl\vert \varepsilon\frac{F(0)}{E(0)}\biggr\vert >K_{1}, $$

then there exists a positive measure set\(\Delta\subset(\mu^{-1},\infty)\)forTso that for\(T\in\Delta\), \(F_{T}\)has a strange attractor Λ admitting no periodic sinks. This is to say that there exists an open neighborhoodUof Λ in\({R}^{m}\)such that\(F_{T}\)has a positive Lyapunov exponent for Lebesgue almost every point inU. Furthermore, \(F_{T}\)admits an ergodic SRB measure, with respect to which almost every point ofUis generic.

In this section we consider the nonlinear delayed differential equation

where \(\mathbf{u}_{t}(\theta) = \mathbf{u}(t+\theta)\), \(\theta\in [-r,0] \) for \(r>0\), \(L_{\mu}:C[-r,0] \to\mathrm{R}^{n}\) is a linear operator, \(f_{\mu}:\mathrm{C}[-r,0] \to\mathrm{R}^{n}\) is a nonlinear term satisfying \(f_{\mu}(\mathbf{0})=\mathbf{0}\), \(D_{\mathbf{u}} f_{\mu}(\mathbf{0})=\mathbf{0}\). \(f_{\mu}\) and \(L_{\mu}\) depend on μ analytically for \(|\mu|\) is sufficiently small, and \(P_{T}=\sum^{\infty} _{n=-\infty}\delta(t-nT)\). When \(\varepsilon=0\), the undisturbed system is

For the linear system \(\dot{\mathbf{u}}=L_{\mu}\mathbf{u}_{t}\), there is an \(n\times n\) matrix \(\eta(\cdot,\mu):[-r,0]\rightarrow \mathrm{R}^{n^{2}}\) such that for any \(\varphi\in\mathrm{C}[-r,0]\)

such that \(a(0)=0\), \(\omega(0)=\omega_{0}>0\), \((d/d\mu)a(0)\neq0\), and there exists \(c>0\) such that for any \(\lambda\in\sigma(\mu)\), \(\lambda\neq\lambda_{1,2}\), \(\operatorname{Re}(\lambda_{i})<-c\), \(i\geq3\).

Then the flow on the central manifold \(\mathrm{W}^{c}\) of system (11)0 can be written by using Hassard’s method [10] in the following normal form:

We further let \(x=\cos\theta\), \(y=\sin\theta\), \(\mathbf{W}=\mathbf{0}\) in (22), then \(\{\hat{\mathbf{s}}_{0}=(\cos \theta,\sin\theta,\mathbf{0})\in S\times\mathcal{B}, \theta\in [0,2\pi)\}\) is the unit circle in \((x,y)\)-plane in \((x,y,\mathbf{W})\)-space. Define

Theorem 2

Let the values ofμandεbe fixed and assume that (a)-(c) hold. Regard the periodTof the forcing as a parameter and define\(F_{T} = F_{\mu,\varepsilon,T}\). Then there exists a constant \(K_{2}\), determined exclusively by\(\phi(\theta)\), such that if

$$\biggl\vert \varepsilon\frac{F(0)}{E(0)}\biggr\vert >K_{2}, $$

then there exists a positive measure set\(\Delta\subset(\mu^{-1},\infty)\)forTso that for\(T\in\Delta\), \(F_{T}\)has a strange attractor Λ admitting no periodic sinks. This is to say that there exists an open neighborhoodUof Λ such that\(F_{T}\)has a positive Lyapunov exponent for Lebesgue almost every point inU. Furthermore, \(F_{T}\)admits an ergodic SRB measure, with respect to which almost every point ofUis generic.

Proof

We can easily see that (B1), (B2) in Theorem 2 correspond to (A1), (A2) in Theorem 1. After transformations, Eq. (23) in \((x,y)\)-plane corresponds to Eq. (10) in \((\xi,\eta)\)-plane, they are on the central manifolds. Obviously the conditions in Theorem 1 are satisfied. □

where\(\tau_{i}\ge0\), \(i=1,2,\ldots,m\), and\(p_{j}^{(i)}\) (\(i=0,1,\ldots,m\); \(j=1,2,\ldots,n\)) are constants. As\((\tau_{1}, \tau_{2},\ldots,\tau_{m})\)vary, the sum of the order of the zeros of\(P(\lambda,e^{ - \lambda\tau_{1} } ,\ldots,e^{ - \lambda\tau_{m} } )\)on the open right half plane can change if and only if there are zeros crossing the imaginary axis.

Let \(\lambda(\tau)=\alpha(\tau)+i\omega(\tau)\) be the root of Eq. (25) near \(\tau=\tau^{(j)}_{k}\) satisfying

When \(E^{*} = E^{*}_{2}, \mbox{or }E^{*}_{3}\), the coefficients of Eq. (25) are \(\alpha _{1}=1-2ac\), \(\alpha_{2}=-2ac-a\), \(\beta_{1}=-2ac\). Under condition (H1), Eq. (25) has at least one positive real root, so \(E^{*}_{2}\) and \(E^{*}_{3}\) are unstable equilibria.

From above, we have obtained the conditions under which \(E_{1}^{*}\) undergoes Hopf bifurcation at \(\tau=\tau_{0}\). Now, we will derive the explicit formulae determining the direction and the stability of the periodic solutions bifurcating from the equilibrium \(E_{1}^{*}\). Let \(\tau=\tau_{0}+\mu\), \(t=\tau\bar{t}\) and omit ‘-’ above t, we rewrite (4)ε as

Assume that \(q(\theta)\) is the eigenvector of \(A(0)\) corresponding to \(i\omega_{0} \tau_{0} \), then \(A(0)q(\theta) = i\tau_{0} \omega _{0} q(\theta)\). It follows from the definition of \(A(0)\) that

In what follows, we will obtain the coordinates to describe the center manifold \(C_{0}\) at \(\mu = 0\). Notice that \(u_{t}(\theta)=(x_{t}(\theta),y_{t}(\theta),z_{t}(\theta))^{T}=zq(\theta )+\bar{z}\bar{q}(\theta)+W(t,\theta)\), then we have

which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value \(\tau_{0}\), i.e., \(\mu_{2}\) determines the directions of a Hopf bifurcation: if \(\operatorname{Re} \{\lambda'(\tau_{0})\}>0\), \(\mu_{2}>0 \) (resp. \(\mu_{2}<0\)), then the Hopf bifurcation is supercritical (resp. subcritical) and the periodic solutions exist for \(\tau>\tau_{0}\) (\(\tau<\tau_{0}\)). If \(\operatorname{Re} \{\lambda'(\tau_{0})\}<0\), however, the bifurcating periodic solutions are on the opposite direction. \(\beta_{2}\) determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if \(\beta_{2}<0\) (\(\beta_{2}>0\)). Furthermore, we can get \(E(0)=\frac{3a^{2}d\tau_{0}\omega^{2}_{0}d_{1}}{d_{1}^{2}+d_{2}^{2}}\), and \(F(0)=\frac{3a^{2}d\tau_{0}\omega^{2}_{0}d_{2}}{d_{1}^{2}+d_{2}^{2}}\). Let \(W=0\), from Eq. (20) and Eq. (22), we have

Thus, if \(3a^{2}d\tau_{0}\omega^{2}_{0}d_{1}>0\), then we have \(E(0)>0\) and \(\phi(\theta)\) is easy to be verified to be a Morse function. So, according to Theorem 2, there is a constant \(K_{2}\) such that if \(\vert \varepsilon\frac{F(0)}{E(0)}\vert >K_{2}\), we can get an observable rank one chaos.

From the discussion of Section 4, we have \(\omega_{0}\doteq0.8805\), \(\tau_{0}\doteq0.04\), \(\operatorname{Re}\{\lambda'{(0)}\}\doteq1.2968\), \(\operatorname{Re} \{c_{1}{(0)}\}\doteq-0.0024\), \(\mu_{2}\doteq0.0018\), \(\beta_{2}\doteq-0.0047\), \(T_{2}\doteq3.6421 \), so we know that the equilibrium \(E_{1}^{*}\) is asymptotically stable when \(\varepsilon=0\) and \(\tau\in{[0,0.04)}\) (see Figure 1). Further, we know that the bifurcation is supercritical and the stable periodic solution emerges from the equilibrium \(E_{1}^{*}\) (see Figure 2).

Figure 1

The trajectories and phase graphs of system (41) with\(\pmb{T=72.3}\),\(\pmb{\varepsilon=0}\),\(\pmb{\tau=0.01}\),\(\pmb{E^{*}}\)is asymptotically stable.

Figure 2

The trajectories and phase graphs of system (41) with\(\pmb{T=72.3}\),\(\pmb{\varepsilon=0}\),\(\pmb{\tau=0.04}\),\(\pmb{E^{*}}\)is unstable, and a stable periodic solution bifurcates from the equilibrium\(\pmb{E^{*}}\).

We also get \(E(0)\doteq0.0024>0\), \(|\frac{F(0)}{E(0)}|\doteq59.7693\). We choose T in interval \((69,73) \), which is quite ‘large’ to afford a long relaxation period between consecutive kicks of the external force, and choose ε in \((0, 1)\). In Figure 3, we present a rank one strange attractor at \((\tau,\varepsilon,T)=(0.04,0.55,69.3)\). In Figure 4, a rank one strange attractor at \((\tau,\varepsilon,T)=(0.04,0.55,72.3)\) is shown. In Figure 5, the largest Lyapunov exponents λ versus ε for system (41) with \(T=69.3\) and \(T=72.3\) are given. The results of our numerical simulations are in perfect agreement with the predictions of the rank one theory in Section 3.

Figure 3

The trajectories and phase graphs of system (41) with\(\pmb{T=69.3}\),\(\pmb{\varepsilon=0.55}\),\(\pmb{\tau=0.04}\), a rank one strange attractor occurs.

Figure 4

The trajectories and phase graphs of system (41) with\(\pmb{T=72.3}\),\(\pmb{\varepsilon=0.55}\),\(\pmb{\tau=0.04}\), a rank one strange attractor occurs.

Figure 5

Largest Lyapunov exponentsλversusεfor system (41) with\(\pmb{T=69.3}\)(left),\(\pmb{T=72.3}\)(right),\(\pmb{\tau=0.04}\)andεvarying from 0 to 1.0.

We have developed rank one theory from an ordinary differential equation to a time-delayed system and considered the existence of rank one chaos in time-delayed Chua’s system. It is shown that rank one strange attractors occur when the delayed system under a periodic kick undergoes supercritical Hopf bifurcation. We also show some results of numerical simulations to support the rank one theory.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Nos. 11061016, 11461036).

Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.